Arithmetic to geometric moments of asset returns
[mg, Cg] = arith2geom(ma,Ca) [mg, Cg] = arith2geom(ma,Ca,t)
 Arithmetic mean of assetreturn data (nvector). 
 Arithmetic covariance of assetreturn data, an

 (Optional) Target period of geometric moments in terms
of periodicity of arithmetic moments with default value 
arith2geom
transforms moments associated
with a simple Brownian motion into equivalent continuously compounded
moments associated with a geometric Brownian motion with a possible
change in periodicity.
[mg,Cg] = arith2geom(ma,Ca,t)
returns mg
,
continuously compounded or "geometric" mean of asset returns over
the target period (nvector), and Cg
, which is
a continuously compounded or "geometric" covariance of asset returns
over the target period (n
byn
matrix).
Arithmetic returns over period t_{A} are modeled as multivariate normal random variables with moments
$$E[\text{X}]={\text{m}}_{A}$$
and
$$\mathrm{cov}(\text{X})={\text{C}}_{A}$$
Geometric returns over period t_{G} are modeled as multivariate lognormal random variables with moments
$$E[\text{Y}]=1+{\text{m}}_{G}$$
$$\mathrm{cov}(\text{Y})={\text{C}}_{G}$$
Given t = t_{G} / t_{A}, the transformation from geometric to arithmetic moments is
$$1+{\text{m}}_{{G}_{i}}=\mathrm{exp}(t{\text{m}}_{{A}_{i}}+\frac{1}{2}t{\text{C}}_{{A}_{ii}})$$
$${\text{C}}_{{G}_{ij}}=(1+{\text{m}}_{{G}_{i}})(1+{\text{m}}_{{G}_{\text{j}}})(\mathrm{exp}(t{\text{C}}_{A}{}_{ij})1)$$
For i,j = 1,..., n.
If t = 1, then Y = exp(X).
This function has no restriction on the input mean ma
but
requires the input covariance Ca
to be a symmetric
positivesemidefinite matrix.
The functions arith2geom
and geom2arith
are
complementary so that, given m
, C
,
and t
, the sequence
[mg,Cg] = arith2geom(m,C,t); [ma,Ca] = geom2arith(mg,Cg,1/t);
yields ma
= m
and Ca
= C
.
Example 1. Given arithmetic
mean m
and covariance C
of monthly
total returns, obtain annual geometric mean mg
and
covariance Cg
. In this case, the output period
(1 year) is 12 times the input period (1 month) so that t
= 12
with
[mg, Cg] = arith2geom(m, C, 12);
Example 2. Given annual arithmetic
mean m
and covariance C
of asset
returns, obtain monthly geometric mean mg
and covariance Cg
.
In this case, the output period (1 month) is 1/12 times the input
period (1 year) so that t = 1/12
with
[mg, Cg] = arith2geom(m, C, 1/12);
Example 3. Given arithmetic
means m
and standard deviations s
of
daily total returns (derived from 260 business days per year), obtain
annualized continuously compounded mean mg
and
standard deviations sg
with
[mg, Cg] = arith2geom(m, diag(s .^2), 260); sg = sqrt(diag(Cg));
Example 4. Given arithmetic
mean m
and covariance C
of monthly
total returns, obtain quarterly continuously compounded return moments.
In this case, the output is 3 of the input periods so that t
= 3
with
[mg, Cg] = arith2geom(m, C, 3);
Example 5. Given arithmetic
mean m
and covariance C
of 1254
observations of daily total returns over a 5year period, obtain annualized
continuously compounded return moments. Since the periodicity of the
arithmetic data is based on 1254 observations for a 5year period,
a 1year period for geometric returns implies a target period of t
= 1254/5
so that
[mg, Cg] = arith2geom(m, C, 1254/5);